Quasimodular forms arising from Jacobi's theta function and special symmetric polynomials

Abstract

Ramanujan derived a sequence of even weight $2n$ quasimodular forms $U_{2n}(q)$ from derivatives of Jacobi's weight $3/2$ theta function. Using the generating function for this sequence, one can construct sequences of quasimodular forms of all nonnegative integer weights with minimal input: a weight 1 modular form and a power series $F(X)$. Using the weight 1 form $\hat{I}¸(q{)}^2$ and $F(X)=\exp (X/2)$, we obtain a sequence $\{Y_n(q)\}$ of weight $n$ quasimodular forms on $\hat{I}{“}_0(4)$ whose symmetric function avatars ${\tilde{Y}}_n(x^k)$ are the symmetric polynomials $T_n(x^k)$ that arise naturally in the study of syzygies of numerical semigroups. With this information, we settle two conjectures about the $T_n(x^k).$ Finally, we note that these polynomials are systematically given in terms of the Borel-Hirzebruch $\hat{A}$-genus for spin manifolds, where one identifies power sum symmetric functions $p_i$ with Pontryagin classes.

15 pages; to appear in Journal of Combinatorial Theory, Series A; edited with minor typos